Logic is the study of consequence. Given a few mathematical statements or facts, we would like to be able to draw some conclusions. For example, if I told you that a particular real-valued function was continuous on the interval \([0,1]\text{,}\) and \(f(0) = -1\) and \(f(1) = 5\text{,}\) can we conclude that there is some point between \([0,1]\) where the graph of the function crosses the \(x\)-axis? Yes, we can, thanks to the Intermediate Value Theorem from Calculus. Can we conclude that there is exactly one point? No. Whenever we find an “answer” in math, we really have a (perhaps hidden) argument. Mathematics is really about proving general statements (like the Intermediate Value Theorem), and this too is done via an argument, usually called a proof. We start with some given conditions, the premises of our argument, and from these we find a consequence of interest, our conclusion.
in-context